1 Modeling maps a physical process to a mathematical representation (e.g. equations) that can be solved. Any physical process is infinitely complex (atom is not the end, there are quarks, …) To be solvable, we have to make approximations. Stating the problem (what we want), and making good approximations (explicitly declared), so it can be solved relatively accurately and efficiently, is the art of modeling. Good model captures the essential features about the particular aspect of the process we would like to know. → Modeling & approximations must be goal-oriented.

2 r  The simplest material model is that of point mass. (m, x): no size or orientation, just mass & position. Earth has size. But Issac Newton ( ) said: consider Earth as a point mass … and Sun as a point mass (drawn as finite so you can see, but conceptually, Zip, Nada, Nothin…)

3 What do we want to know? The orbital period… the length of a year. Not just Earth, also Mars, Venus… Moon… comets… eclipses Newton wanted to unravel the astronomical mysteries that took thousands of Sumerian, Egyptian, Maya… priests hundreds of years to “figure out” in one stroke. Just plug in different parameters (remarkably few) into the same equation. Such is the power of a physical model. Basic ingredients: m(d 2 x/dt 2 ) = F(x) -- main equation F(x) = -GmM(x/|x| 3 ) -- “constitutive law” G = × kg -1 m 3 s universal constant M, m -- material parameters x, t: -- variables

4 The traffic flow problem Step 1. Define scope & goals of the model: To understand traffic pattern on one-way road with no entrance/exit: - What is the maximum carrying capacity of the road? - How long does it take to travel between two points on the road? - If there is traffic jam, how does it move, and how long to clear? Step 2. Make conceptual sketch and define basic quantities: x x 3, v 3, L 3 x 4, v 4, L 4 x 5, v 5, L 5 x 2, v 2, L 2 x 1, v 1, L 1 aperture of width D and centered at x N(D,x,t): number of vehicles in [x-D/2, x+D/2) at time t.  (x,t) ≡ N(D, x, t) / D is the density v(x,t): the average velocity  (x,t) and v(x,t) are coarse-grained, continuous fields. realistic D ~ 600ft if vehicle spacing 30ft agent-based description coarse-grained, field description

5 Step 3. Develop a mathematical description of the conceptual sketch: The molecular dynamics approach: Write down evolution equation for discrete degrees of freedom: {x i (t)} → v i (t) = f(…, x i-2 (t), x i-1 (t), x i (t), x i+1 (t), x i+2 (t), …) for every i → Update: …, x i (t+  t)=x i (t)+v i (t)  t, … → {x i (t+  t)} Repeat and this forms an autonomous loop (a set of ODEs). In general, one could also put in randomness. The continuum, or coarse-grained, approach: The degree of freedom (DoF) are continuous fields such as  (x,t), v(x,t) Q(x,t) ≡  (x,t)v(x,t) is the vehicle flux Step 4. Write down equations that govern the variables: xx realistic  x ~ 1000ft Q(x,t)Q(x,t) Q(x+  x,t) x x+  x vehicles in – vehicles out = vehicles accumulated Q(x,t)dt - Q(x+  x,t)dt =  (x,t+dt)  x -  (x,t)  x

6 Step 5. Introduce constitutive relations between the variables: Intuitively we can appreciate that there should be some relationship between v and . Your driving experience differs greatly in a bumper-to-bumper jam at 5PM versus an empty lane at 3AM! We make the following general observations about v(  ): 1. v(  =0) is still finite. 2. v(  decreases monotonically with increasing . 3. There exists a  max where v(  max ) ≈ 0. (1 mile = 5280 feet)

7 The simplest v(  ) relation that satisfies the above is: v(  ) = v max (1-  /  max ) Let’s see where this assumption leads us… (modeling…) Q(  )=  v(  )=  v max (1-  /  max )=  max v max (  /  max )(1-  /  max ) 01  /  max v()v()Q()Q() v max Q max Q max is only  max v max /4! Why when there is a fire, people shouldn’t rush to the exit…  max /2 is the critical density above which a stampede could happen. Step 6. Reduce the governing equations by approximations: None. “downward spiral” or catastrophe

8 Step 7. Non-dimensionalize the equations by scaling the variables:

9 Step 8. Solve the dimensionless equations to infer the behavior of the system: x f (x,0)=f 0 (x) f(x,1) f(x,2)

10 c is therefore identified as the wave speed in the linear ODE problem. What is the significance of the linear ODE solution in the context of the nonlinear ODE problem? Consider a homogeneous traffic flow:  *(x,t)=  * 0 This is a solution to the nonlinear PDE. Now consider a small disturbance in traffic flow pattern:  *(x,t) =  * 0 +  *(x,t)

11 Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations (Cambridge University Press, 1996).

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14 c(  * 0 ) is therefore the propagation speed of an infinitesimal disturbance on the originally homogeneous flow pattern of  *(x,t)=  * 0. -v max Q max 01  /  max v()v()c()c() v max 0 0 disturbance pattern stationary disturbance pattern travels backward disturbance pattern travels forward, but with slower speed than the average vehicle speed.

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17 Press, Teukolsky, Vetterling, Flannery, Numerical recipes: the art of scientific computing, 3rd ed. (Cambridge University Press, 2007)

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